In this chapter, we turn to the problem of completing the set of equations presented in Section 4.8 by introducing specific constitutive equations. They are material-specific and thus depend upon the constitution of the material.

We close this book with a brief discussion of one of the most important and challenging unsolved problems in the mechanics of fluids: turbulence. As it remains as much descriptive art as predictive science, it is appropriate to call upon visual and poetic sources for inspiration to examine this daunting subject. In the visual realm, the subject has been illustrated with a well-known sketch from da Vinci, seen in Fig. 14.1.

Here we consider some standard problems in multi-dimensional viscous flow. As for one-dimensional viscous flow, application areas are widespread and can include ordinary pipe flows as well as microscale fluid mechanics. We will restrict attention to problems that are steady and laminar. Most of the problems will be incompressible, except for one dealing with a problem in natural convection, Section 11.2.6, and another in compressible boundary layers, Section 11.2.7.

In this chapter, we introduce the complex numbers system – an extension of the well-known real numbers. Complex numbers arise naturally in many problems in mathematics and science and allow us to study polynomial equations that may not have real solutions (such as x2 + 5 = 0). As we will see, many familiar algebraic properties remain valid in the complex number system. In particular, we show that the complex numbers form a field and that the quadratic formula and the triangle inequality can still be used in this new number system.

Linear Algebra is a branch of mathematics that deals with linear equation, systems of linear equations, and their representation as functions between algebraic structures called vector spaces. Linear Algebra is an essential tool in many disciplines such as engineering, statistics, and computer science and is also central within mathematics, in areas such as analysis and geometry. Much like the definition of a field, a vector space is defined through a list of axioms, motivated by concrete observations in familiar spaces, such as the standard two-dimensional plane and the three-dimensional space. We begin this chapter by taking a closer look at real n-dimensional spaces and vectors and then move on to discussing abstract vector spaces and linear maps. Our experience with sets and functions, developed in previous chapters, as well as certain proof techniques, will prove to be useful in our discussion.

We begin our journey by taking a closer look at some familiar notions, such as quadratic equations and inequalities. And, rather than using mechanical computations and algorithms, we focus on more fundamental questions: Where does the quadratic formula come from and how can we prove it? What are the rules that can be used with inequalities, and how can we justify them? These questions will lead us to looking at a few proofs and mathematical arguments. We highlight some of the main features of a mathematical proof, and discuss the process of constructing mathematical proofs. We also review informally the types of numbers often used in mathematics and introduce relevant terminology.

In this chapter, we introduce and discuss fundamental notions in mathematics: sets, functions, and axioms. Sets and functions show up everywhere in mathematics and science and are common tools used in mathematical arguments. Moreover, proving statements about sets and functions can further develop our proof-writing and communication skills. We also demonstrate, in Section 2.3, how axioms are used in mathematics as initial assumptions, from which other statements can be derived.

In this chapter, we take a step back to discuss, more generally, the language of mathematics and some proof techniques and strategies. In the previous chapters, we have seen numerous mathematical notions, theorems, proofs, and examples. As you have probably noticed, communicating mathematical arguments and ideas in a coherent and precise way is at the core of the subject.

This chapter is devoted to studying, in more depth, the set of integers Z, its structure, and properties. The integers play a fundamental role in many areas of mathematics, science, and beyond. The integers are closely related to the set of natural numbers and thus are often used in problems involving counting, sequences, and structures with finitely many elements (such as finite fields).

Given two infinite sets, is there a sensible way to decide which one is larger? For instance, if A is the set of even integers, and B is the interval [0, 1], is there a way to compare their sizes? In this section, we focus on such questions and introduce the notion of cardinality, which is used to describe “how many elements” a (potentially infinite) set has. This leads to some interesting and counterintuitive consequences. However, we must first prepare the ground by discussing injections, surjections, bijections, and related results.

In this chapter, we discuss relations, a central notion in mathematics. As we will shortly see, we have already encountered many mathematical relations without using this terminology. We begin by formally defining what a relation is and then introduce a special type of relation – equivalence relations and the associated notion of an equivalence classes. In Section 7.4, we study an important and useful equivalence relation: congruence modulo n.

Real analysis is a branch of mathematics focusing on the study of real numbers and related objects. Sets of real numbers, sequences, functions, and series of real numbers are at the core of the subject. The notions of limits and convergence are central in analysis and are used to investigate such objects. Learning real analysis means, in part, deepening our understanding and studying the theoretical foundations of calculus topics. For these reasons, many view real analysis as a rigorous version of calculus. In this chapter, we look at how limits of sequences and function can be formally defined. The precise definitions may require some effort to grasp, but it is absolutely essential for advanced studies in mathematics and related fields. Formal definitions of limits allow us to not only prove various statements (such as the Extreme and the Intermediate Value Theorems, often proved in a Real Analysis course), but also investigate more complicated functions and sequences. Our experience with proof writing and logical statements will be invaluable for our discussion. We also highlight the use of limits to defining continuity and differentiability of functions.